The survey, conducted by The Dallas Morning News and the University of Texas at Tyler, shows 45 per cent of registered voters in the state would vote for Mr McConaughey.
Only 33 per cent would vote for Governor Abbott, and 22 per cent would vote for someone else.
Mr McConaughey has long-teased a run for the governor role of his home state. Though buoyed by name recognition, his double-digit poll lead over the incumbent Republican is one of the first signs that he could be a serious contender.
When broken down by party, results of the poll show a divide in enthusiasm for the star of The Dallas Buyers Club and A Time to Kill.
Only 30 per cent of Republicans chose Mr McConaughey, while 56 per cent opted for Governor Abbott.
Two-thirds of Democrats favoured him over the current governor, with only 8 per cent choosing Mr Abbott. Independents split 44 per cent to 28 per cent for the actor.
Mr McConaughey has criticised both parties in the past and would likely run as a moderate, making him an awkward fit for either party.
Some 51 per cent of Democrats would want a more progressive candidate over a centrist, and only 20 per cent of Republicans want a more moderate candidate for their party.
The majority of conservative, evangelical, and retirement age voters in the Republican party preferred Mr Abbott over a centrist candidate, and they are more likely to vote in any primary election.
While overall 47 per cent of Republicans opted for Mr Abbott, 14 per cent wanted a more conservative candidate, and 18 per cent wanted someone more like Donald Trump.
Political scientist Mark Owens, who conducted the poll, noted that some of the party base who supported Mr Trump do like an outsider celebrity candidate who campaigns against corruption and a need for change.
When combined with those who want a more moderate figure in the governor’s mansion, there could be a path for Mr McConaughey in a party primary.
“Matthew McConaughey gets a huge boost from tremendous name recognition, and recognition for what he does to help Texans and add to the celebration of the state’s successes,” Mr Owens said.
“Most of our survey respondents know his story, but many are waiting to see how he opens his next chapter.”
Governor Abbott has managed to maintain the support of Texas Republicans and independent voters during his seven years in office.
It is unclear what impact the handling of Covid-19 and the massive power outage during February’s winter storm will have on Mr Abbott’s political future.
The same poll shows a narrow plurality of Texans distrust him to protect them from the pandemic (47 per cent to 46 per cent), while 54 per cent had a favourable opinion of the governor’s response to the winter storm utility problems versus 44 per cent who did not.
Conducted between 6 to 13 April, the poll surveyed 1,126 registered voters and has a margin of sampling error of plus or minus 2.92 percentage points.
Mr McConaughey has said in the past that he is giving “honest consideration” to running and described politics as a “broken business” in need of an “aggressively centric” approach.